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Jonathan Brundan

Publications and source records attributed to Jonathan Brundan.

At least 37 records · Page 2Linked to original sources
arXiv2019

Whittaker coinvariants for $\mathrm{GL}(m|n)$

Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$. In this paper we study the {\em Whittaker coinvariants functor}, which is an exact functor from category $\mathcal O$ for $\mathfrak{gl}_{m|n}(\mathbb{C})$ to a certain category of finite-dimensional modules over $W_{m|n}$. We show that this functor has properties similar to Soergel's functor $\mathbb V$ in the setting of category $\mathcal O$ for a semisimple Lie algebra. We also use it to compute the center of $W_{m|n}$ explicitly, and deduce some consequences for the classification of blocks of $\mathcal O$ up to Morita/derived equivalence.

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arXiv2018

On the definition of Heisenberg category

We revisit the definition of the Heisenberg category of central charge k. For central charge -1, this category was introduced originally by Khovanov, but with some additional cyclicity relations which we show here are unnecessary. For other negative central charges, the definition is due to Mackaay and Savage, also with some redundant relations, while central charge zero recovers the affine oriented Brauer category of Brundan, Comes, Nash and Reynolds. We also discuss cyclotomic quotients.

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arXiv2018

A basis theorem for the degenerate affine oriented Brauer-Clifford supercategory

We introduce the oriented Brauer-Clifford and degenerate affine oriented Brauer-Clifford supercategories. These are diagrammatically defined monoidal supercategories which provide combinatorial models for certain natural monoidal supercategories of supermodules and endosuperfunctors, respectively, for the Lie superalgebras of type Q. Our main results are basis theorems for these diagram supercategories. We also discuss connections and applications to the representation theory of the Lie superalgebra of type Q.

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arXiv2017

Representations of the oriented skein category

The oriented skein category $OS(z,t)$ is a ribbon category which underpins the definition of the HOMFLY-PT invariant of an oriented link, in the same way that the Temperley-Lieb category underpins the Jones polynomial. In this article, we develop its representation theory using a highest weight theory approach. This allows us to determine the Grothendieck ring of its additive Karoubi envelope for all possible choices of parameters, including the (already well-known) semisimple case, and all non-semisimple situations. Then we construct a graded lift of $OS(z,t)$ by realizing it as a 2-representation of a Kac-Moody 2-category. We also discuss the degenerate analog of $OS(z,t)$, which is the oriented Brauer category $OB(δ)$.

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arXiv2017

The p-centre of Yangians and shifted Yangians

We study the Yangian $Y_n$ associated to the general linear Lie algebra $\mathfrak{gl}_n$ over a field of positive characteristic, as well as its shifted analog $Y_n(σ)$. Our main result gives a description of the centre of $Y_n(σ)$: it is a polynomial algebra generated by its Harish-Chandra centre (which lifts the centre in characteristic zero) together with a large $p$-centre. Moreover, $Y_n(σ)$ is free as a module over its center. In future work, it will be seen that every reduced enveloping algebra $U_χ(\mathfrak{gl}_n)$ is Morita equivalent to a quotient of an appropriate choice of shifted Yangian, and so our results will have applications in classical representation theory.

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arXiv2017

Super Kac-Moody 2-categories

We introduce generalizations of Kac-Moody 2-categories in which the quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced by the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka.

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arXiv2017

Type C blocks of super category O

We show that the blocks of category O for the Lie superalgebra q_n associated to half-integral weights carry the structure of a tensor product categorification for the infinite rank Kac-Moody algebra of type C. This allows us to prove two conjectures formulated by Cheng, Kwon and Lam. We then focus on the full subcategory consisting of finite-dimensional representations, which we show is a highest weight category with blocks that are Morita equivalent to certain generalized Khovanov arc algebras.

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arXiv2017

Monoidal supercategories

In the literature, one finds several competing notions for the super (i.e., Z/2-graded) analog of a monoidal category. The goal of this paper is to clarify these definitions and the connections between them. We also discuss in detail the example of the odd Temperley-Lieb supercategory. In a forthcoming article, we will exploit the formalism developed here in order to define super analogs of the Kac-Moody 2-categories of Khovanov-Lauda and Rouquier.

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arXiv2016

Type A blocks of super category O

We show that every integral block of category O for the general linear Lie superalgebra is equivalent to a corresponding block of category O for the queer Lie superalgebra. This implies the truth of the Kazhdan-Lusztig conjecture for the so-called type A blocks of category O for the queer Lie superalgebra, as formulated by Cheng, Kwon and Wang.

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arXiv2016

Categorical actions and crystals

This is an expository article developing some aspects of the theory of categorical actions of Kac-Moody algebras in the spirit of works of Chuang-Rouquier, Khovanov-Lauda, Webster, and many others.

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arXiv2016

Tensor product categorifications and the super Kazhdan-Lusztig conjecture

We give a new proof of the "super Kazhdan-Lusztig conjecture" for the Lie super algebra $\mathfrak{gl}_{n|m}(\mathbb{C})$ as formulated originally by the first author. We also prove for the first time that any integral block of category O for $\mathfrak{gl}_{n|m}(\mathbb{C})$ (and also all of its parabolic analogs) possesses a graded version which is Koszul. Our approach depends crucially on an application of the uniqueness of tensor product categorifications established recently by the second two authors.

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arXiv2014

A basis theorem for the affine oriented Brauer category and its cyclotomic quotients

The affine oriented Brauer category is a monoidal category obtained from the oriented Brauer category (= the free symmetric monoidal category generated by a single object and its dual) by adjoining a polynomial generator subject to appropriate relations. In this article, we prove a basis theorem for the morphism spaces in this category, as well as for all of its cyclotomic quotients.

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arXiv2013

Quiver Hecke algebras and categorification

This is a brief introduction to the quiver Hecke algebras of Khovanov, Lauda and Rouquier, emphasizing their application to the categorification of quantum groups. The text is based on lectures given by the author at the ICRA workshop in Bielefeld in August, 2012.

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arXiv2013

Homological properties of finite type Khovanov-Lauda-Rouquier algebras

We give an algebraic construction of standard modules (infinite dimensional modules categorifying the PBW basis of the underlying quantized enveloping algebra) for Khovanov-Lauda-Rouquier algebras in all finite types. This allows us to prove in an elementary way that these algebras satisfy the homological properties of an `affine quasi-hereditary algebra.' In simply-laced types these properties were established originally by Kato via a geometric approach. We also construct some Koszul-like projective resolutions of standard modules corresponding to multiplicity-free positive roots.

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arXiv2012

Principal W-algebras for GL(m|n)

We consider the (finite) $W$-algebra $W_{m|n}$ attached to the principal nilpotent orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb C)$. Our main result gives an explicit description of $W_{m|n}$ as a certain truncation of a shifted version of the Yangian $Y(\mathfrak{gl}_{1|1})$. We also show that $W_{m|n}$ admits a triangular decomposition and construct its irreducible representations.

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arXiv2011

Gradings on walled Brauer algebras and Khovanov's arc algebra

We introduce some graded versions of the walled Brauer algebra, working over a field of characteristic zero. This allows us to prove that the walled Brauer algebra is Morita equivalent to an idempotent truncation of a certain infinite dimensional version of Khovanov's arc algebra, as suggested by recent work of Cox and De Visscher. We deduce that the walled Brauer algebra is Koszul whenever its defining parameter is non-zero.

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arXiv2011

Moeglin's theorem and Goldie rank polynomials in Cartan type A

We use the theory of finite W-algebras associated to nilpotent orbits in the Lie algebra g = gl_N(C) to give another proof of Moeglin's theorem about completely prime primitive ideals in the enveloping algebra U(g). We also make some new observations about Joseph's Goldie rank polynomials in Cartan type A.

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